Xu Zongben

Uncertainty measures of roughness of knowledge and rough sets in incomplete information systems

In this paper we address uncertainty measures of roughness of knowledge and rough sets by introducing rough entropy in incomplete information systems. We make only one assumption about unknown values: the real value of a missing attribute is one from the attribute domain. However, we do not assume which one. We prove that the rough entropy of knowledge and the rough entropy of rough sets decrease monotonously as the granularity of information grows smaller through finer partitionings. These conclusions are helpful to understand the essence of rough set theory and essential to seek new efficient algorithm of knowledge reduction in incomplete information systems.

A problem of D.H. Lehmer and its mean square value formula

Abstract Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1⩽b bc≡a( mod q) . Let N(a,q) denote the number of solutions of the congruence equation bc≡a( mod q) with 1⩽b,c ∑′ a=1 q (N(a,q)− 1 2 φ(q)) 2 , and give a sharp asymptotic formula.

On the uniform continuity of metric projections in Banach spaces

Let M be a convex Chebyshev subset of a uniformly convex and uniformly smooth Banach space. It is proved that the metric projection PM of X onto M is uniformly continuous on every bounded subset of X. Moreover, a global and explicit estimate on the modulus of continuity of the metric projection is obtained.

Ball algorithms for constructing solutions of nonlinear operator equations

Abstract In the most general setting we extend the previous theory of the region contraction algorithm developed recently by several authors for solving nonlinear operator equations. It is demonstrated that the extended algorithm still possesses properties such as generating error bounds automatically, providing a computational test for the existence of solutions and being globally convergent under nominal conditions. As example of applications, we specialize the extended theory in particular to nonlinear operator equations of monotone type.

Generalized theory and some specializations of the region contraction algorithm I - Ball operation

We describe a new algorithm named Region Contraction Algorithm for solving certain nonlinear equations, and establish the convergence of the algorithm and give an error estimation. It is shown that this general theory includes all of present existing ball iterations as special cases.

Sign matrix, regular splittings and monotonic enclosure of solutions for nonlinear system of equations, part II

AbstractAs a continuation of part I of the paper under the same title, we develop general monotonic enclosure methods for the couple systems of the splitting equations

Multichannel synthetic aperture radar imaging method

\left\{ {\begin{array}{*{20}c} {x = G\left( {\left[ x \right]_a ,\left[ x \right]_b ,\left[ y \right]_c } \right),} \\ {y = G\left( {\left[ y \right]_a ,\left[ y \right]_b ,\left[ x \right]_c } \right),} \\ \end{array} } \right.

On Selection of the Observation Model for Multilook Application of Sparse Microwave Imaging

which models the system of equations associated with hybrid and asynchronous monotonicity as well as convexity. The resulting algorithms and convergence theorems generalize and unify various known methods and monotonic enclosure theorems established by other authors.